带有止步和状态相依的M/Ej/1/N排队系统的矩阵解法

研究了带有止步和服务率依赖于状态的M/Ej/1/N排队系统.顾客到达系统时,以一定的概率选择进入系统或止步(不进入系统).顾客接受服务的服务率依赖于系统中的顾客数,当系统中的顾客数不超过临界值k时,服务员慢速服务;否则,服务员快速服务.利用分块矩阵的方法,推出了稳态概率向量所满足的矩阵形式的迭代公式,给出了稳态概率的表达式和计算过程.作为特例,考虑了N=4时系统稳态概率的计算.在此基础上,还求出了系统的一些性能指标,并建立了以临界值k为控制变量的费用模型.通过数值分析,求出了使费用函数最小的最优临界值k*,并进一步研究了模型参数对最优临界值和最优费用的影响.

Matrix Solution of the State-Dependent M/Ej/1/N Queue with Balking

We consider an M/Ej/1/N queuing system with balking and state-dependent service rate.If a customer on arrival finds other customers in the system,it either decides to enter the queue or balks with a constant probability.Customers are served with two different rates depending on the number of customers in the system.When the number of customers in the system is less than or equal to the critical value k,the server has slow service rate,otherwise the server has fast service rate.By using the block matrix technology,we deduce the matrix form iterative formula of the steady-state probability vectors, we get the explicit expression of the steady-state probabilities and present a algorithm for calculating the steady-state probabilities,especially we give the explicit steady-state probabilities for n=4.In addition,we get the expression of the system performance measures,and develop a cost model to determine the optimal critical value k to minimize the total expected cost per unit time.We investigate the impact of some parameters on the optimal cost and the optimal critical value.